Reduction of Presymplectic Manifolds with Symmetry

TL;DR

This paper develops a unified reduction method for presymplectic manifolds with symmetry, capable of removing gauge and rigid symmetries simultaneously, with applications to dynamical systems and field theories.

Contribution

It introduces a comprehensive reduction approach for presymplectic systems that handles both gauge and non-gauge symmetries at once, improving upon existing step-by-step methods.

Findings

Unified reduction method for presymplectic manifolds with symmetry.

Application to time-dependent dynamical systems and gauge theories.

Comparison with other reduction procedures.

Abstract

Actions of Lie groups on presymplectic manifolds are analyzed, introducing the suitable comomentum and momentum maps. The subsequent theory of reduction of presymplectic dynamical systems with symmetry is studied. In this way, we give a method of reduction which enables us to remove gauge symmetries as well as non-gauge ``rigid'' symmetries at once. This method is compared with other step-by-step reduction procedures. As particular examples in this framework, we discuss the reduction of time-dependent dynamical systems with symmetry, the reduction of a mechanical model of field theories with gauge and non-gauge symmetries, and the gauge reduction of the system made of a conformal particle.